Wave Lengths of Periodic Waves for the Vakhnenko Equation

GUO Li-na; CHEN Ai-yong; HUANG Wen-tao

doi:10.21656/1000-0887.370020

Volume 37 Issue 7

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GUO Li-na, CHEN Ai-yong, HUANG Wen-tao. Wave Lengths of Periodic Waves for the Vakhnenko Equation[J]. Applied Mathematics and Mechanics, 2016, 37(7): 678-690. doi: 10.21656/1000-0887.370020

Citation:

GUO Li-na, CHEN Ai-yong, HUANG Wen-tao. Wave Lengths of Periodic Waves for the Vakhnenko Equation[J]. Applied Mathematics and Mechanics, 2016, 37(7): 678-690. doi: 10.21656/1000-0887.370020

Citation:

GUO Li-na, CHEN Ai-yong, HUANG Wen-tao. Wave Lengths of Periodic Waves for the Vakhnenko Equation[J]. Applied Mathematics and Mechanics, 2016, 37(7): 678-690. doi: 10.21656/1000-0887.370020

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Wave Lengths of Periodic Waves for the Vakhnenko Equation

doi: 10.21656/1000-0887.370020

¹School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, P.R.China;²Department of Mathematics, College of Science,Hezhou University, Hezhou, Guangxi 542800, P.R.China

Funds: The National Natural Science Foundation of China(11361017)

Received Date: 2016-01-13
Rev Recd Date: 2016-01-25
Publish Date: 2016-07-15

Abstract

Abstract

The wave lengths of smooth periodic traveling wave solutions to the Vakhnenko equation were studied. The Vakhnenko equation was reduced to a planar polynomial differential system through the transformation of variables. The polynomial differential system was treated with the critical period bifurcation method based on the dynamical system theory. The main results involve the monotonicity properties of periodic function T(h) (or wave length function λ(a)). In comparison with the wave length for the KdV equation, wave length function λ(a) monotonically decreases to a finite value rather than monotonically increases to infinity. This shows that, for fixed wave speed c, there exist no smooth periodic wave solutions with arbitrarily small wave lengths or arbitrarily large wave lengths, to the Vakhnenko equation.
- Vakhnenko equation,
- periodic wave,
- periodic function,
- wave length,
- monotonicity

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References(20)

References

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